Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants

This is an abridged version of our Habilitation thesis. In these notes, we aim to summarize our research interests and achievements as well as motivate what drives our work: symmetry, structure and invariants. The paradigmatic example which permeates and often inspires our research is the Weyl algebra ${\mathbb{A}}_1$.